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The Paulus–Rozenfeld–Thompson Graph on 26 Vertices Revisited and Related Combinatorial Structures

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Isomorphisms, Symmetry and Computations in Algebraic Graph Theory (WAGT 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 305))

Abstract

This paper deals with the strongly regular graph T, having parameters (26, 10, 3, 4) and the largest possible automorphism group \(G=\mathrm {Aut}(T)\) of order 120. The group G in its action on the vertex set of T has two orbits of length 20 and 6. Many special features of the graph T and its group G make it a remarkable object in algebraic graph theory. The presentation, arranged in the style of a tutorial, describes the graph T from many viewpoints of mathematics and computer algebra. Special attention is paid to the links of T with such classical structures as the Petersen graph, semi-icosahedron, icosahedron, dodecahedron, two-graphs on 26 points, Paley graphs, inversive plane and generalized quadrangles of order 5 and others.

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Acknowledgements

The authors gratefully acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic (VEGA) under the grant 1/0988/16. MK gratefully recalls his communication with late Jaap Seidel and Ernie Shult, which was crucial for the shaping of his vision of the theory of two-graphs. He is also much obliged to all late and alive members of the former Moscow group for a fruitful communication. Special thanks go in this direction to Igor Faradžev, the creator of the program part of COCO. Support of Christian Pech, Sven Reichard, as well as of Dennis Epple was very helpful and efficient. We are much obliged to Andries Brouwer, Peter Cameron and Gareth Jones for communications, which were supporting spirit of this project. We thank also Rosemary Bailey, Katie Brodhead, Willem Haemers, Josef Lauri, Mikhail Muzychuk, Andy Woldar and Yaokun Wu for helpful remarks. Personal attention of Don Thompson was crucial for the start of the project. The initial impulse in the creation of the paper came from Sasha Ivanov, Ján Karabáš, Roman Nedela, Akihiro Munemasa, all our partners in the planning and running the event WL 2018. Finally, the authors are pleased to thank Jozef Širáň for his attention to this paper, as well as an anonymous referee for a very enthusiastic positive evaluation.

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Authors and Affiliations

  1. Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, 81005, Bratislava, Slovak Republic

    Štefan Gyürki

  2. Department of Mathematics, Ben-Gurion University of the Negev, 8410501, Beer Sheva, Israel

    Mikhail Klin & Matan Ziv-Av

  3. Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, 97401, Banská Bystrica, Slovakia

    Mikhail Klin

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  1. Štefan Gyürki
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Correspondence to Mikhail Klin .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, University of Southampton, Southampton, UK

    Gareth A. Jones

  2. Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia

    Ilia Ponomarenko

  3. Department of Mathematics and Statistics, The Open University, Milton Keynes, UK

    Jozef Širáň

Supplements

Supplements

See Tables 15, 16, 17 and 18

Table 15 130 circles in \(\mathcal M(5)\)
Full size table
Table 16 Structure constants of \(\mathcal M\): the element in the jth row and kth column in the matrix \(P_i\) corresponds to constant \(p_{i,j}^k\)
Full size table
Table 17 Main data about subgroups of \(\mathrm {Aut}(T)\)
Full size table
Table 18 List of generators appearing in Table 17
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Gyürki, Š., Klin, M., Ziv-Av, M. (2020). The Paulus–Rozenfeld–Thompson Graph on 26 Vertices Revisited and Related Combinatorial Structures. In: Jones, G., Ponomarenko, I., Širáň, J. (eds) Isomorphisms, Symmetry and Computations in Algebraic Graph Theory. WAGT 2016. Springer Proceedings in Mathematics & Statistics, vol 305. Springer, Cham. https://doi.org/10.1007/978-3-030-32808-5_4

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